In The Road to Reality, § 15.2, Roger Penrose introduces the concept of a fibre bundle. Here I'll modify his notation to that of Wikipedia and other sources, so that B stands for base space (rather than the bundle), F for fibre, and E for the total space, which I think Penrose calls the (fibre) bundle itself. His first example is the real line bundle over S1. This is the product space S1 x R1, where S1 is a circle (any circle?) and R1 is the 1-dimensional vector space of real numbers over the reals with the natural definition of scalar multiplication. A product space is called a trivial bundle of F over B. So far so good. He identifies this structure with a (the?) cylinder. I'm having trouble with his next example, which he calls a twisted bundle, which he identifies with a (the?) Möbius strip. There are pictures of Möbius strips. I can see what twisted means when it refers to a strip of paper, but what does twisted mean when it refers to a fibre bundle? I think I follow what a (cross-)section means, at least when the bundle is trivial: a function s : B --> E such that the projection [itex]p \; \circ \; s(x) = x[/itex]. (This according to Wikipedia's more formal intro; Penrose's cross-section is only the image of this map, or can be "thought of as" the image; he's a little vague here.) At least, I think can see how this works for a product B x F. But how is it different in the twisted case? Penrose, having just described the case for a trivial bundle B x F: "This is like the ordinary idea of the graph of a function [...] More generally, for a twisted bundle B, any cross-section of B defines a notion of a 'twisted function' that is more general than the ordinary idea of a function." In Fig. 15.7, he shows a curve drawn in a loop around a cylinder, and one around a Möbius strip. I think the idea is that the loop can't go all the way around the Möbius strip and join up with itself (and thus can't be continuous) unless it crosses the central line around the strip (which line Penrose identifies with the zero section). I suppose what's puzzling me is that these copies of F are disjoint, so how can one copy of, say, the vector space (R1, R1, regular multiplication) be said to have something analogous to an orientation or direction compared to another? Whereabouts in the definition of the Möbius strip bundle does this idea come in?